Systems we can look at tend to be in stable equilibrium for a very good reason: they stay in stable equilibrium- that's the definition of stable equilibrium! A system in a non-equilibrium or unstable equilibrium state will move away from it very rapidly. Again, that's the definition of "non-equilibrium" and "unstable equilibrium". Those systems that "stand still" for us to study are the ones that are in stable equilibrium.

The fact that stable equilibrium corresponds to minimal energy is due pretty much to the fact that energy is defined so as to make that so.